Dynamic pricing and investment for fashion products under reference effect

In this study, we discuss on the dynamic pricing and investment strategy for fashion products which decay in value, such as fashion clothes. The products are sold by an enterprise under a monopoly market in a finite horizon. Due to the obsolescence of fashion products, their fashion level decreases over time. Obviously, the selling price has a negative effect on demand, while the fashion level has a positive effect. Moreover, we introduce a reference value, which is used to indicate consumers' requirements for fashion level. We assume that investment on products performance or goodwill can reduce consumers' requirements for fashion level, an optimal control model is established. To maximize the enterprise's total profit, we obtain the trajectories of the optimal pricing and investment strategy by utilizing the Pontryagin's maximum principle. The optimal strategy is to gradually lower the selling price and then slightly raise it at the end of the sales period and reduce investment as products become obsolete. Afterwards, we provide a numerical example to illustrate our conclusions. Through the sensitivity analysis of the main parameters, we capture the influence of the parameters on an optimal decision. Moreover, some management implications are given.